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Properties of Inequalities: A B A 5 A 5 A 5

Inequalities follow specific rules: < means less than, > means greater than, ≤ means less than or equal to, and ≥ means greater than or equal to. The rules include: adding or subtracting a number preserves the inequality; multiplying or dividing by a positive number preserves it but a negative number reverses it; taking the negative or reciprocal reverses the inequality if both numbers are positive or negative; and if a<b and b<c then a<c. Inequalities can be combined and properties such as square roots or powers are used to compare values.

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242 views2 pages

Properties of Inequalities: A B A 5 A 5 A 5

Inequalities follow specific rules: < means less than, > means greater than, ≤ means less than or equal to, and ≥ means greater than or equal to. The rules include: adding or subtracting a number preserves the inequality; multiplying or dividing by a positive number preserves it but a negative number reverses it; taking the negative or reciprocal reverses the inequality if both numbers are positive or negative; and if a<b and b<c then a<c. Inequalities can be combined and properties such as square roots or powers are used to compare values.

Uploaded by

Textile Academia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Properties of Inequalities

Compiled by: Mustafizur Rahman Rumon

Inequality problem deals with numbers that are less than, greater than, or equal
to other numbers. The following rules apply to all inequalities:

< means less than, thus 3 < 4

> means greater than, thus 5 > 2

≤ means less than or equal to, thus 3 ≤ 4 and 3 ≤ 3

≥ means greater than or equal to, thus 5 ≥ 2 and 2 ≥ 2

• For any numbers a and b: a> b means that a-b is positive.


• For any numbers a and b: a< b means that a-b is negative.
•a< 5 means a is less than 5. a can be 4,3,2,1,-1,-2,……….
•a> 5 means a is greater than 5. a can be 5,6,7,8,9,……….
•a ≤ 5 means a is less than or equal 5. a can be 5,4.9, 4.8, 4.5, 4, 1, 0, -1,-2,….
•a ≥ 5 means a is greater than or equal 5. a can be 5, 5.1, 5.2, 6, 7, 8, 9,…….
• Adding a number to an inequality or subtracting a number from the inequality
preserves the inequality.
If a< b, then a+ c< b+c and a−c <b−c .
Example: 3<7 ⟹ 3+100<7+ 100⟹ 103<107
3<7 ⟹ 3−100< 7−100 ⟹−97<−93

• Adding inequalities in the same direction preserves them.


If a< b∧c< d , t h en a+c <b+ d .
Example:3<7∧5<10 ⟹ 3+5<7+10 ⟹ 8<17.

• Multiplying or dividing an inequality by a positive number preserves the


inequality.
a b
If a< b , and c is positive, then ac <bc and c < c
Example: 3<7 ⟹ 3 ×100<7 ×100 ⟹ 300<700
3 7
3<7 ⟹ 3 ÷100< 7 ÷100 ⟹ <
100 100
• Multiplying or dividing an inequality by a negative number reverses the
inequality.
a b
If a< b , and c is negative, then ac >bc and c > c
Example: 3<7 ⟹ 3 × (−100 ) >7 × (−100 ) ⟹−300>−700
3 −7
3<7 ⟹ 3 ÷ (−100 )> 7 ÷ (−100 ) ⟹− >
100 100

• Taking negative reverses an inequality.


If a< b , t h en−a>−b ,∧if a> b , t h en−a<−b .
Example: 3<7 ⟹−3>−7 ,∧7>3 ⟹−7<−3.

• If two numbers are each positive or each negative, taking reciprocals reverses
an inequality.
1 1
If a and b are both positive or both negative and a< b , t h en a > b .
1 1 1 1
Examples:3<7 ⟹ 3 > 7 ∧−7<−3 ⟹ −7 > −3

• If a< b∧b<c ,t h en a< c.


Example:3<5∧5<7 , t h en 3<7.

• If 0< x <1 ,∧ais positive ,then xa<a .


For example: 0.85 ×5<5

• If 0< x <1 ,∧m∧n are integers with m> n>1 ,then xm < x n < x .
For example:¿

• If 0< x <1 ,then √ x> x .


3 3
For example:
√ >
4 4

1 1
• If 0< x <1 , then x > x . In fact, x >1.
1
For example: 0.2 >1>0.2.

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